Geometric and Material Nonlinear Analysis of Frames with Nonuniform Cross-Sections

在結構物的後挫屈反應中，由於多條鄰近平衡路徑之出現，常會造成數個彎曲路徑的產生，而這些鄰近的路徑往往使得非線性迭代過程，難以收斂到所要的路徑上。有鑒於此，本文將對所使用的增量迭代法中的一些關鍵步驟作探討，期能有助於程式的模擬與結構物後挫屈反應的追蹤。在使用有限元素分析方法時，所謂的校正階段指的是由元素位移來計算元素的內力，由於影響到不平衡力的計算，因此內力計算要愈精確愈好；所謂的預測階段則是指在給定的載重增量下計算結構物的位移，基本上此一階段屬於線性化的假設，因此只要不影響迭代解的收斂，近似到某個程度是可以接受的。而一個好的增量迭代法，必須具備以下三個要求，包括通過極限點時的數值穩定性、能夠自動調整載重增量大小，與自動調整載重方向。在本文中，一些高度複雜的後挫屈問題將用來作為驗證。 在材料非線性分析中，本文採用Attalla等(1994)所提出的擬塑鉸法。由於此法使用全量式表示，需要使用數值積分與有限微分法來求得彈塑性柔度矩陣係數，因此在電腦運算量上是很費時的。為了克服以上的缺點，Leu與其研究團隊把擬塑鉸法做了一些改進，使得彈塑性柔度係數可以直接用簡單的表示式求得。在本文中，除了對擬塑鉸法中的等斷面與變斷面公式作探討，也將對複雜冗長的變斷面公式作修正，並且提出一個簡化的變斷面方法。最後，將採用三個標準的平面鋼結構彈塑性問題，來驗證與比較等斷面、變斷面公式，以及所提出的變斷面方法之可靠性。The postbuckling response of structures with multi winding loops is characterized by the appearance of multi adjacent equilibrium paths, which often makes the iterations difficult to converge to the desired path. In this study, some key issues for tracing the postbuckling response of a structure using an incremental-iterative approach are discussed. Concerning the finite element equations used, it is essential that the corrector used for recovering the element forces from the element displacements be made as accurate as possible, and that the predictor for computing the structural displacements under given load increments, which are approximate by nature due to linearization involved, be accurate to the level not to misguide the direction of iterations. As for the incremental-iterative scheme, it is required to be: (1) numerically stable in passing the limit points, (2) self-adjustable for the load increments, and (3) automatic in reversing the loading direction. To demonstrate the ideas involved, some examples with highly complicated post-buckling responses will be solved in this study. Referring to the material nonlinear analysis, the quasi-plastic-hinge approach proposed by Attalla et al. (1994) is adopted in this study. This approach is formulated using the total form expressions, in which numerical integrations and the finite difference method are employed to obtain the inelastic flexibility coefficients. Such a procedure is computationally inefficient in general. To overcome the above drawback, Leu and his co-workers made some improvements in the quasi-plastic-hinge approach, in which the elastic-plastic flexibility coefficients were determined explicitly. In this study, the formulation of the quasi-plastic-hinge approach for uniform and nonuniform cross sections will be conducted. To reduce to complexity involved in the formulation for nonuniform cross sections, an improved approach is proposed. To compare different formulations for the inelastic analysis of frames with uniform and nonuniform cross sections, including the proposed approach, three benchmark problems of planar steel structure will be used as the basis and thoroughly studied. By the numerical studies, the reliability of the quasi-plastic-hinge approach for uniform and nonuniform cross sections will be verified.1 Introduction 1 1.1 Motivation and background 1 1.2 Objective 4 2 Key Considerations in Tracing the Postbuckling Response of Structures with Multi Winding Loops 7 2.1 Introduction 7 2.2 Updated Lagrangian formulation 11 2.3 Predictor stage in incremental-iterative nonlinear analysis 13 2.3.1 Truss element 14 2.3.2 Two-dimensional beam element 18 2.4 Corrector stage in incremental-iterative nonlinear analysis 20 2.4.1 Truss element 20 2.4.2 Two-dimensional beam element 21 2.5 Path-tracing scheme for the postbuckling response of structures 24 2.5.1 Incremental-iterative nonlinear analysis 24 2.5.2 Generalized displacement control method 26 2.5.3 Characteristics of the generalized stiffness parameter 27 2.5.4 Algorithm for incremental-iterative nonlinear analysis 28 3 Numerical Verification on Geometric Nonlinear Analysis 35 3.1 Verification of the GDC method by using different formulations in the predictor and corrector stages 35 3.1.1 Two-member truss 35 3.2 Verification of the GDC method for three-dimensional trusses 37 3.2.1 12-member truss 37 3.2.2 24-member shallow dome 37 3.3 Verification of the GDC method for planar frames 39 3.3.1 Williams’ toggle 39 3.3.2 Shallow arch 39 3.3.3 Hinged circular arch 40 3.3.4 Axially-compressed cantilever 41 3.3.5 Cantilever under shear load 42 3.3.6 Pinned-fixed square diamond frame 42 4 Explicit Inelastic Stiffness Matrix for Beam-Column Elements with Uniform and Nonuniform Cross-Sections 69 4.1 Introduction 69 4.2 Moment-Curvature Relationship Based Method 72 4.2.1 Beam-column element: total form formulation 72 4.2.2 The elastic-plastic stiffness matrix and tangent stiffness matrix 75 4.3 Beam-column element: incremental form formulation for uniform cross sections 77 4.4 Beam-column element: incremental form formulation for nonuniform cross sections 81 4.5 Beam-column element: proposed incremental form formulation for non- uniform cross sections 84 4.6 Moment-Curvature Relationship from Curve Fitting 87 4.6.1 The initial yield surface and full plastification surface 87 4.6.2 Curve-Fitted Moment-Curvature Relationship 91 4.6.3 Strategy for the computer program implementation 93 5 Numerical Studies on Material Nonlinear Analysis 107 5.1 Verification of the procedure for uniform cross sections 107 5.1.1 El-Zanaty portal frame 107 5.1.2 Kanchanalai leaned frame 108 5.1.3 Kanchanalai portal frame 108 5.2 Verification of the procedure for nonuniform cross sections 109 5.2.1 El-Zanaty portal frame with columns of nonuniform cross sections 110 5.2.2 Kanchanalai leaned frame with columns of nonuniform cross sections 111 5.2.3 Kanchanalai portal frame with columns of nonuniform cross sections 112 5.2.4 L-shaped frame with a column of nonuniform cross section 112 5.3 Verification of the proposed method 113 5.3.1 El-Zanaty portal frame with columns of nonuniform cross sections 113 5.3.2 Kanchanalai leaned frame with columns of nonuniform cross sections 113 5.3.3 Kanchanalai portal frame with columns of nonuniform cross sections 114 5.3.4 L-shaped frame with a column of nonuniform cross section 114 5.4 Further study of the proposed method 115 5.4.1 Influence of different numbers of inner segments 115 5.4.2 Influence of the number of elements and the number of inner segments used to represent each element 116 6 Conclusions 145 6.1 Summary 145 6.2 Recommendations for Future Work 147 Reference 14